Balanced Synaptic Input Shapes the Correlation betweenNeural Spike TrainsAshok Litwin-Kumar1,2*, Anne-Marie M. Oswald2,3, Nathaniel N. Urban2,3, Brent Doiron2,4*

1 Program for Neural Computation, Carnegie Mellon University and University of Pittsburgh, Pittsburgh, Pennsylvania, United States of America, 2 Center for the Neural

Basis of Cognition, Pittsburgh, Pennsylvania, United States of America, 3 Department of Biology, Carnegie Mellon University, Pittsburgh, Pennsylvania, United States of

America, 4 Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania, United States of America

Abstract

Stimulus properties, attention, and behavioral context influence correlations between the spike times produced by a pair ofneurons. However, the biophysical mechanisms that modulate these correlations are poorly understood. With a combinedtheoretical and experimental approach, we show that the rate of balanced excitatory and inhibitory synaptic inputmodulates the magnitude and timescale of pairwise spike train correlation. High rate synaptic inputs promote spike timesynchrony rather than long timescale spike rate correlations, while low rate synaptic inputs produce opposite results. Thiscorrelation shaping is due to a combination of enhanced high frequency input transfer and reduced firing rate gain in thehigh input rate state compared to the low state. Our study extends neural modulation from single neuron responses topopulation activity, a necessary step in understanding how the dynamics and processing of neural activity change acrossdistinct brain states.

Citation: Litwin-Kumar A, Oswald A-MM, Urban NN, Doiron B (2011) Balanced Synaptic Input Shapes the Correlation between Neural Spike Trains. PLoS ComputBiol 7(12): e1002305. doi:10.1371/journal.pcbi.1002305

Editor: Olaf Sporns, Indiana University, United States of America

Received August 8, 2011; Accepted October 30, 2011; Published December 22, 2011

Copyright: � 2011 Litwin-Kumar et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: A.L.K. is supported by the Department of Defense NDSEG Program. A.M.M.O. and N.N.U. are supported by Grant R01 DC0005798 (to N.N.U.) from theNational Institute of Deafness and Other Communication Disorders. B.D. is supported by National Science Foundation grant DMS-0817141 and is a Sloan ResearchFellow. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: alk@cmu.edu (ALK); bdoiron@pitt.edu (BD)

Introduction

Correlations between the spike trains of neuron pairs are

observed throughout the central nervous system [1]. The

correlation between a pair of neurons’ spike trains can change

depending on the state of their neural circuit. For instance,

correlated neural activity is altered by stimulus properties [2,3],

anesthetics [4,5], stimulus adaptation [6], focus of spatial attention

[7,8], and the behavioral context of a task [9]. The level of spike

train correlation between neuron pairs has implications for the

accuracy of population codes [10], the formation of neural

assemblies [11], and the propagation of neural activity [12,13].

Nonetheless, only recently has attention been given to the

mechanisms by which correlated activity is modulated

[14,15,16,17,18,19,20].

Cortical neurons receive a mixture of excitatory and inhibitory

synaptic inputs, resulting in spiking activity that is driven by input

uctuations rather than the input mean [21,22]. This state is often

described as balanced, to denote that the mean excitatory and

inhibitory inputs that neurons receive are approximately equal

[23,24]. Balanced activity is inuenced by stimulus properties and

history [25,21], as well as internal brain state [26]. These changes

can modulate the integration properties of single neurons, strongly

inuencing neuronal activity [22]. For example, increases in the

firing rate of balanced pre-synaptic activity afferent to a neuron

can reduce single neuron firing rate gain [27,28,29,30,31,32].

Further, an increase in the temporal correlation between the

arrival times of excitatory pre-synaptic inputs increases the firing

rate of a post-synaptic target neuron [33,34,35], while correlations

between excitatory and inhibitory inputs can reduce output

activity [34,36]. The impact of such shifts in the temporal

structure of synaptic input is amplified when the post-synaptic cell

has a small integration timescale, as expected for neurons in the

high input rate, balanced state [22]. These examples deal with

synaptic activity convergent to a single target cell. However, what

is less studied is the role that the balanced state plays in modulating

the responses of a pair of neurons subject to a common synaptic

input. In this study, we consider this latter scenario and show that

shifts in balanced pre-synaptic population activity modulate the

magnitude and timescale of the correlations of spike trains from

pairs of post-synaptic neurons.

We first explore a model system and show that output spike

train correlations from a pair of neurons are modulated by varying

the rate of uctuating, balanced excitatory and inhibitory inputs.

Specifically, we demonstrate that an increase synaptic input rate

leads to an increase of short-timescale output correlation (i.e.

precise spike synchrony) while correlation at long timescales (i.e

firing rate co-variation) remains unaffected, or even decreases.

Due to the differential affects of our mechanism on short and long

timescale spiking activity we label the combined modulation

correlation shaping. Correlation shaping has been observed in various

sensory systems [2,3,37,38], yet the core mechanisms underlying

the modulation remain unknown. We present linear response

analysis showing that the enhancement of output synchrony

through an increase of input rate results from a shift in single

neuron integration properties that favors the transfer of high

frequency inputs. Dynamic clamp recordings from cortical

neurons verify our theoretical predictions. Finally, in a feedfor-

PLoS Computational Biology | www.ploscompbiol.org 1 December 2011 | Volume 7 | Issue 12 | e1002305

ward network model, we show how correlation shaping supports a

selective propagation of network responses, so that activity can be

gated by correlations in complex neuronal networks. In total, our

work extends mechanisms of single neuron firing rate control

include the control of pairwise correlations, thereby providing a

bridge between single neuron and network state modulation.

Methods

Conductance-Based Neuron ModelWe modeled neurons as leaky integrate-and-fire units receiving

conductance input [39]. Each neuron had an intrinsic timescale

t~20 ms and leak reversal potential EL~{65 mV. Excitatory

and inhibitory synaptic input caused conductance changes ge(t)and gi(t) with reversal potentials Ee~0 mV and Ei~{75 mV so

that the membrane potential dynamics followed:

dV

dt~

1

t(EL{V )z

ge(t)

C(Ee{V )z

gi(t)

C(Ei{V ):

When V reached a threshold voltage Vth~{55 mV, the neuron

spiked and the voltage was reset to Vre~{65 mV.

We modeled the excitatory and inhibitory synaptic conduc-

tances as Poisson processes with rates Re and Ri consisting of series

of d-functions with heights ae~:01 and ai~:02. This framework

was used for all of the simulations presented and provides a

minimal model that captures our main results (for simulations of

other models, see Supplementary Figures). These inputs consisted

of independent processes private to each neuron as well as a

shared component presynaptic to all neurons, yielding

Re=i~cRse=iz(1{c)Ri

e=i where superscripts i and s denote

independent and shared components, respectively. For large rates,

this input was approximated as a diffusion process

[39,40,41,42](Figure S1):

ae=i(Ee=i{V )X

n

d(t{tne=i)&ae=i(Ee=i{V ) Re=iz

ffiffiffiffiffiffiffiffiRe=i

qje=i(t)

� �,

where je=i(t)~ffiffiffiffiffiffiffiffiffiffi1{cp

jie=i(t)z

ffiffifficp

jse=i(t) was a Gaussian white

noise process with unit intensity. This allowed us to write our

voltage equation in the form

dV

dt~

1

teff

(Eeff {V )zs(V )j(t), ð1Þ

where teff :t

1ztaeReztaiRi

, Eeff :ELztReaeEeztRiaiEi

1ztReaeztRiai

,

and s2(V)~a2eRe(Ee{V )2za2

i Ri(Ei{V )2. Note that as the

rates of excitation and inhibition Re and Ri increase in a balanced

manner, teff decreases, s increases, and Eeff does not change

substantially because of the excitation and inhibition balance.

For our simulations and calculations, we set s(V )~s(Eeff ). This

approximation ignored the multiplicative nature of the noise, which

in our simulations did not substantially change the results (Figure

S1), since the change in teff and s(Eeff ) were sufficient to modulate

neuronal responses. To simulate pairs of neurons receiving

correlated input, we set the fluctuating input to each neuron to be

s(V )j(t)~s(Eeff )(ffiffifficp

js(t)zffiffiffiffiffiffiffiffiffiffi1{cp

ji(t)), ð2Þ

where js(t) was shared across both neurons while ji(t) was

independent for each neuron. We note that, although the

correlation in output spike trains depended on the degree of pre-

synaptic overlap, Eq. (2) shows that s(Eeff ), and hence the firing

rate of neurons in our model, was independent of c. The rate of

excitatory input in the low state was 1.50 kHz and 6.16 kHz in the

high state, with the inhibitory rate chosen to elicit a firing rate of

15 Hz in both cases. Simulations were performed using an Euler-

Maruyama numerical integration scheme with a simulation time-

step of 0.005 ms.

Solving for Transfer Function and Power Spectrum withFokker-Planck Techniques

We next developed a theoretical framework to study the

behavior of the above system and compared our theory against

simulations of the stochastic system. For completeness, we write

the governing equations used to calculate the single neuron power

spectrum CCii(f ) and transfer function AA(f ); these techniques are

fully presented in [42] and we refer the reader there for further

details. Letting h(V)~1

teff

(Eeff {V), the voltage distribution

P(V ,t) associated with the stochastic differential equation (1) obeys

the Fokker-Planck equation:

LP

Lt~{

LJ

LV~{

LLV

h(V )P½ �z 1

2

LLV

s2 LP

LV

� �,

where J(V ,t) is the probability flux [43]. The boundary conditions

for the probability distribution and flux at threshold are P(Vth)~0and J(Vth,t)~n(t), where n(t) is the firing rate. Furthermore, the

flux obeys J(V ,t)~n(t) for V[½Vre,Vth� and is 0 otherwise.

For time independent Eeff and s the steady state distribution

P0(V ) obeys:

LP0

LV~{

2

s2J0{h Vð ÞP0½ �,

LJ0

LV~n0d V{Vreð Þ{n0d V{Vthð Þ:

Using the normalization conditionÐVth

{? P0(V)dV~1, we can

solve for the steady state firing rate n0.

Author Summary

Neurons in sensory, motor, and cognitive regions of thenervous system integrate synaptic input and output trainsof action potentials (spikes). A critical feature of neuralcomputation is the ability for neurons to modulate theirspike train response to a given input, allowing task contextor past history to affect the flow of information in thebrain. The mechanisms that modulate the input-outputtransfer of single neurons have received significantattention. However, neural computation involves thecoordinated activity of populations of neurons, and themechanisms that modulate the correlation between spiketrains from pairs of neurons are relatively unexplored. Weshow that the level of excitatory and inhibitory input that aneuron receives modulates not only the sensitivity of asingle neuron’s response to input, but also the magnitudeand timescale of correlated spiking activity of pairs ofneurons receiving a common synaptic drive. Thus, whilemodulatory synaptic activity has been traditionally studiedfrom a single neuron perspective, it can also shape thecoordinated activity of a population of neurons.

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 2 December 2011 | Volume 7 | Issue 12 | e1002305

In order to study the system’s response to a correlated,

fluctuating input, it is necessary to study the system’s response to

time-dependent inputs. This is done most effectively by writing a

time-dependent Fokker-Planck equation in the Fourier domain:

LPP

LV~{

2

s2JJ{h Vð ÞPP� �

,

LJJ

LV~{2pif PP{nn fð Þd V{Vthð Þze{2pif d V{Vreð Þ,

where (XX ) denotes the Fourier transform of X and nn(f ) is

computed with initial condition V~Vre. Solving this equation

yields the Fourier transform of the first passage time density hh(f )

[42]. The power spectrum CCii(f )~n0 1z2<½gg(f )�ð Þ, where gg(f ) is

calculated from the well known renewal relation

gg(f )~hh(f )=(1{hh(f )) [44].

Finally, we compute the transfer function AA(f ). Suppose that we

add a time-varying periodic current I(t)~I0e2pift to the right hand

side of Eq (1). If we let I0 be sufficiently small, we can compute the

spike train response to these time-dependent modulations.

Decomposing the probability density, flux, and firing rate into

steady state and modulated components:

P~P0zPI e2pit, J~J0zJI e2pift, r~r0zAe2pift,

and then solving the Fokker-Planck equation for the time-

dependent terms, we obtain a new set of equations:

{LPPI

LV~

2

s2 Vð Þ JJIzh Vð ÞzI0P0

� �,

{LJJI

LV~ivPPI{rrQe{ivd V{Vreð Þ,

with boundary conditions

PPI (Vth)~0, JJI (Vth)~AA:

These equations were solved numerically [42] obtaining a solution

for the transfer function AA(f ).

Experimental TechniquesSurgery: Somatosensory (S1) cortical slices were prepared from

CBJ/Bl6 mice age P19-26. All surgical procedures followed the

guidelines approved by the Carnegie Mellon Animal Welfare

Committee. The mice were anesthetized with isoflourane and

decapitated. The brain was exposed, removed from the skull and

immersed, in ice cold oxygenated (95%O2{5%CO2) ACSF (in

mM: 125 NaCl, 2.5 KCl, 25 NaHCO3, 1.25 NaH2PO4, 1.0

MgCl2, 25 Dextrose, 2 CaCl2) (all chemicals from Sigma, USA).

Coronal slices (300 mm) of barrel cortex made using a vibratome

(Leica, Place). The slices were maintained in ACSF at 370C for

30 min then rested at room temperature (20{220C) for 1 hr prior

to recording (31{350C).

Electrophysiology: L2/3 pyramidal neurons were visualized

using infrared-differential interference contrast microscopy (Olym-

pus, Center Valley, PA). Whole cell, dynamic clamp recordings

were performed using a MultiClamp 700B amplifier (Molecular

Devices, Union City, CA). Data were low pass filtered (4 kHz) and

digitized at 50 kHz using an ITC-18 (Instrutech, Mineola, NY)

controlled by custom dynamic clamp software (R. Gerkin; http://

rick.gerk.in/software/recording-artist/) written in IgorPro (Wave-

metrics, Lake Oswego, OR). Pipettes were pulled from borosilicate

glass (2.0 mm, outer diameter) on a Flaming/Brown micropipette

puller (Sutter Instruments, Novato, CA) to a resistance of 6–

10 MV. The intracellular solution consisted of (in mM) 130 K-

gluconate, 5 KCl, 2 MgCl2, 4 ATP-Mg, 0.3 GTP, 10 HEPES,

and 10 phosphocreatine.

Stimulation: Pyramidal cells (n = 8) were directly stimulated by a

series (50–100 trials) of simulated noisy synaptic currents in dynamic

clamp. Each trial was 4 s in duration with a 5 s inter-trial interval;

the period of rest was used to ensure that stability of the recordings.

For each trial, excitatory (Ee: 0 mV) or inhibitory (Ei: 260 mV)

synaptic conductance inputs were simulated as Poisson distributed

spike times convolved with alpha function ge,i(t)~�gge,it

te,ie1{t=te,i .

(�gge~1 nS, �ggi~1 nS, te~6 ms, ti~8 ms). The Poisson rates for

excitatory and inhibitory inputs were equal to one another

(Re~Ri), and were set to 3 kHz in the low state and 7.5 kHz in

the high state. These rates were higher than in the simulations to

ensure high spike time variability, since the input variability is

attenuated by the finite temporal extent of the synaptic timescales.

For each state, half of these inputs were common to all neurons

stimulated and half were newly generated on each trial for every

neuron. This produced an input correlation, c, of 0.5 between any

given pair of neurons. This setup permitted 8:(8{1)=2~28pairwise comparisons. Since the synaptic drive was subthreshold,

a bias current (0.3–0.7 nA) was added such that the balanced

conductance fluctuations produced a mean cortical firing rate of (4–

6 Hz) in both the low and high states.

Feedforward NetworkWe studied a layered network in which a population of 100

leaky integrate-and-fire neurons (Layer 2) received balanced input

from a pre-synaptic layer (Layer 1) with c~:2 and provided

excitatory input to two distinct downstream targets. Neurons in

Layer 1 were assumed to be Poisson as in previous sections, and

the total input to a Layer 2 neuron was therefore approximated by

a diffusion process. In particular, the voltage dynamics of each

Layer 2 neuron followed Eqs. 1 and 2.

The downstream target was also modeled as leaky integrate-

and-fire neuron. Because we wished to fix the timescale of the

downstream target, we assumed delta-function, current-based

synapses so that the voltage V of the downstream neuron followed:

dV

dt~

1

tdownstream

(EL{V )zad

X100

c~1

Xk

d(t{tkc )

where c~1 . . . 100 indexes the neurons in Layer 2 and k indexes

the spikes in each Layer 2 neurons’ spike train. We compared

tdownstream~20 ms and tdownstream~3 ms. For tdownstream~20 ms,

we set ad~0:12 mV and for tdownstream~3 ms, ad~0:4 mV so

that the neurons fired at comparable rates given identical input.

Other parameters, including leak, threshold, and reset voltages

were identical to the model previously studied.

Results

Modulation of Correlation SusceptibilityIn general, it is difficult to determine the specific changes in a

neural system’s dynamics that cause changes in spike train

correlations. We studied a framework in which common inputs

drive the correlations between the spike trains of a pair of neurons

[45,46,47]. If the degree of input correlation, c, is small, a linear

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 3 December 2011 | Volume 7 | Issue 12 | e1002305

approximation relating c to the output spike correlation, r, is

written as:

r&Sc:

Here the quantity S, termed the correlation susceptibility, determines

the extent to which two neurons’ spike trains will be correlated given

a fixed level of correlation between the inputs they receive [17].

Throughout this study, we focused on a pair of neurons that

shift their output correlation (r1?r2) due to a change in their pre-

synaptic drive (Figure 1A). Under our linear model, two simple

explanations for the shift in output correlation are possible. First,

the shift may simply reflect a change in the correlation of the

inputs that the neuron pair receives (c1?c2; Figure 1B). While this

answer appears straightforward, understanding shifts in input

correlation requires detailed anatomical knowledge of the network

architecture, in the absence of which simplifying assumptions are

required [48].

A second explanation for the shift in output correlation is a shift in

correlation susceptibility (S1?S2), even when the input correlation

remains fixed (Figure 1C). Because S relates the correlations in the

spiking output of neurons to their common input, we expect S to be

sensitive to how each neuron integrates its input. Indeed, single

neuron response properties such as firing rate and neural excitability

determine the extent to which neurons become synchronized by

shared input [49,17,18,19,20]. There has been substantial work on

how single neuron properties, such as firing rates, are modulated

[27,28,29,30,31,32,50,51,52,53,54],suggesting that S should also be

open to modulation. We focused on this second mechanism and

established how modulations of single neuron responses also

modulated pairwise correlations in cortical populations.

Low and High Rate Synaptic Input StatesWe first investigated the transfer of input correlations to output

spike train correlations in a simplified two-neuron network. Each

neuron received conductance-based, pre-synaptic inputs from a

mixed population of excitatory and inhibitory neurons (Figure 2A).

To model the stochastic nature of cortical activity, the arrival times

of both excitation and inhibition were modeled as Poisson

processes. We set the relative strengths and rates of excitation

and inhibition so that the mean input was balanced [23,24], and

the average membrane potential was below spiking threshold.

Balanced pre-synaptic activity results in large membrane fluctu-

ations that trigger spikes in a random, aperiodic pattern, consistent

with in vivo recordings from cortical neurons [21,22].

Shifts in the activity level of a recurrent cortical population are

observed in many neural systems and have been shown to affect

the response properties of neurons in vitro and in vitro [22,55,31].

To explore the modulatory effects of balanced synaptic input, we

considered the neuron model in two states: a low state, in which

pre-synaptic input arrived at a low rate, and a high state, in which

pre-synaptic input arrived at a high rate (Figure 2A). While the

level of balanced fluctuations may lie on a continuum, we

compared two representative points, analogous to high and low

activity states in a cortical network [56,26]. A clear consequence of

the shift from low to high states was an increase in the variability of

both the input current and membrane potential response, due to

greater fluctuating input (Figure 2B). This increase of input

variability was reflected in an increase in spiking variability, with

the coefficient of variation of the inter-spike intervals increasing

from 0.73 in the low state to 0.91 in the high state. A second

consequence of an increase in pre-synaptic rate was the reduction

of the membrane time constant t (Figure 2B). This was expected,

since the membrane time constant t*C=g, with C the membrane

capacitance and g the total membrane conductance [57]. As g is

roughly proportional to the pre-synaptic rates, an increase in the

rate of synaptic input lead to a decrease in t. Taken together, the

shift from the low to high state evoked a more stochastic and faster

membrane potential response.

We first examined the effect of balanced synaptic input on firing

rate gain, the slope of the firing rate curve when plotted as a

function of excitatory input strength. When the rate of balanced

excitatory and inhibitory synaptic input changed from low to high,

the neuron’s firing rate gain was substantially reduced (Figure 2C).

This gain decrease in the high background state has been studied

extensively in theoretical and in vitro work [27,28,29,30,31,32] as

well in vivo under specific stimuli conditiona [31]. In the high

state,larger membrane potential fluctuations increased firing rates

for weak inputs. However, there was also a decrease of the net

membrane input resistance, causing an increase in the rheobase

current (minimum steady current required to recruit spiking). The

combination of these two effects lead to an overall reduction in

firing rate gain [29]. We next explored the consequences of gain

modulation via balanced activity for correlation transfer by pairs

of neurons.

Correlation Shaping with Synaptic ActivityTo study the effects of balanced excitatory and inhibitory inputs

on pairwise spike train correlations, we extended our model to

include a pair of post-synaptic neurons receiving overlapping pre-

Figure 1. Mechanisms of correlation modulation. (A) The spike train correlation between a pair of neurons shifts from r1 to r2 as the state ofpre-synaptic field shifts. (B) Mapping between input correlation c and output correlation r. The change in output correlation in panel A may be dueto a change in input correlation from c1 in state 1 to c2 in state 2. (C) An alternative mechanism by which output correlation can change is that thecorrelation susceptibility S changes from S1 in state 1 to S2 in state 2, with input correlation c fixed throughout.doi:10.1371/journal.pcbi.1002305.g001

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 4 December 2011 | Volume 7 | Issue 12 | e1002305

synaptic inputs (Figure 3A). Previous work has shown that the

output firing rate affects correlation susceptibility [17]. To

preclude any firing rate-induced effects, the synaptic input was

adjusted so that the average output firing rate of each neuron

remained at 15 Hz in low and high states (Figure 2C). Further-

more, there was a fixed overlap in the input populations, so that

the input correlation also remained constant in both network states

(Figure 3A). Thus, any change in the output spike train correlation

induced by changing synaptic input will be due exclusively to a

shift in correlation susceptibility (Figure 1C).

We found that the timescale over which the two spike trains

were correlated was dependent on the level of balanced synaptic

activity (Figure 3A, Right). When the synaptic rate increased from

the low to high state, the magnitude of the peak of the cross-

correlation function near zero lag increased, reflecting greater

spike time synchrony between the neurons. However, this increase

was not present for longer lags, and the spike train cross-

correlation function was unchanged or reduced for sufficiently

long lags (w10 ms).

To quantify this change in output correlation over a range of

timescales, we first counted the number of spikes nT1 and nT

2 that

the two neurons emitted in intervals of T milliseconds. We next

computed the spike count correlation as a function of window size:

rT~Cov(nT

1 ,nT2 )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var(nT1 )Var(nT

2 )q , ð3Þ

where Cov and Var denote covariance and variance, respectively.

In the framework of our simple circuit (Figure 3A), correlation in

output spike trains rT was a consequence of a shared input

correlation c. For small c, linear response theory [17] takes the

output correlation to be a linear function of the input correlation

(Figures 1B,C; 3B):

rT~ST c: ð4Þ

In our model, this linear relationship held for a range of c, in both

low and high states and at both short and long T (Figure 3B).

Further, the rT values produced were, in magnitude, consistent

with in vivo recordings from a variety of systems [58,3,2,6]. When

comparing rT for the low and high states at fixed c, a differential

change of correlation at different timescales was evident.

Specifically, rlowT vrhigh

T for small T (Figure 3B, T = 3 ms), while

rlowT wrhigh

T for large T (Figure 3B, T = 50 ms). This differential

modulation of correlation occured over a broad range of

timescales, with rlowT and rhigh

T intersecting only once (Figure 3C),

and we label the modulation a shaping of correlation [38]. This

substantial change in both the magnitude and timescale of

correlation must involve a nontrivial change in how the neurons

process their inputs, since the input correlation c and firing rate

were the same in both low and high states. We note that the

qualitative results of our study are also valid for larger c (Figure S2)

and different synaptic strengths (Figure S3).

Since rT?0 as T?0 [59], changes in rT at small T are

necessarily smaller in magnitude. However, synchrony at short

timescales can have large effects on downstream targets sensitive to

coincident pre-synaptic spikes [12] and indeed the peak of the

cross-correlation function increased substantially in the high state

(Figure 3A, Right). To properly compare correlation shaping at

small and large T we considered the ratio ShighT =Slow

T ~rhighT =rlow

T ,

providing a relative measure across the low and high states. The

ratio was a decreasing function of T , with substantial changes in

correlation at both short and long timescales (Figure 3D). The

negative slope of the curve indicates that increases in the rate of

balanced synaptic activity favor spike synchronization rather than

long timescale correlation. Finally, the spectral measure of spike

train coherence between the two spike trains in both states

exhibited a decrease for low frequencies but a significant increase

for high frequencies in the high state (Figure 3E). Here, the

increase for high frequencies, which occurs over a broad range of

frequency space, is related to the increase in short timescale

synchrony, consistent with the spike count correlation shaping.

Correlation shaping is an unexpected feature of balanced

synaptic activity. For subthreshold membrane potential dynamics

(or any other linear system) the ratio rhighT =rlow

T is equal to 1 for all

T assuming a fixed input correlation (Figure 3D, gray line). The

mechanism that shapes correlation transfer so to promote spike

train synchronization over long timescale correlation in the high

state (Figure 3D) is the focus of the next section.

Figure 2. Single cell statistics in low and high synaptic input states. (A) Left: Schematic of low (top) and high (bottom) states. Excitatoryinputs occurred at a rate of 1.5 kHz in the low state and 6.2 kHz in the high state, modeling the activity from a pool of pre-synaptic cells. Inhibitoryinputs were chosen so that output firing rates were fixed at 15 Hz in both states. Right: Synaptic inputs converged onto a conductance-based leakyintegrate-and-fire neuron model. Sample membrane potential traces of the neuron model in both the low (top) or high (bottom) states are shown.The total input current in either state is plotted below each membrane potential trace. (B) The input current variability (s) and membrane potentialtime constant (t) for both the low (top) and high (bottom) input states. (C) Firing rate of a neuron as the level of excitatory input is varied, showingdecreased gain in the high input state compared to the low input state. The balanced condition in both low and high states resulted in an outputfiring rate of 15 Hz. Solid lines were calculated using our theory (see Methods). Dots correspond to numerical simulations of the model system.Standard error is smaller than the width of the dots.doi:10.1371/journal.pcbi.1002305.g002

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 5 December 2011 | Volume 7 | Issue 12 | e1002305

Relationship between Correlation Susceptibility andNeuronal Integration

Correlation shaping is a property of the joint statistics of a pair

of neurons. However, since the input correlation was the same in

the low and high states of our model, then the mechanism

underlying the shaping is hypothesized to be related to changes in

single neuron input integration and spike emission across the two

synaptic states (Figure 1C rather than 1B). In this section, we show

that correlation shaping is a consequence of a shift in the single

neurons’ frequency response across the low to the high input state.

The spike train auto-correlation Cii(t) and cross-correlation

Ci=j(t) functions are written as:

Cij(t)~

ð?{?

yi(t)yj(t{t)dt{�yyi�yyj , ð5Þ

where yi(t)~P

i d(t{tik), with tik labeling the kth spike time

from neuron i (i~1,2). Here �yyi is the mean firing rate of neuron i.

We are interested in the joint spike count correlation for the

neuron pair, where the spike count for neuron i over a window of

length T is nTi ~

Ð T

0yi(t)dt (we take the neuron’s stochastic

dynamics to be in statistical equilibrium). The spike count variance

and covariance are related to integrals of auto- and cross-

correlation functions [44], yielding an alternate expresion for rT :

rT~Cov(nT

1 ,nT2 )ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Var(nT1 )Var(nT

2 )q ~

Ð T

{TCij(t)(T{jtj)dtÐ T

{TCii(t)(T{jtj)dt

ð6Þ

In the second equality we have, for simplicity, assumed that

C11(t)~C22(t) (or equivalently Var(nT1 )~Var(nT

2 )). These inte-

Figure 3. Pairwise cell statistics in low and high rate synaptic input regimes. (A) Schematic of low (left) and high (center) states with samplemembrane traces. The marginal statistics of both cells are as reported in Figure 2, with a fixed overlap of excitatory and inhibitory pre-synaptic inputsfor the cell pair. The input correlation is c~0:5 for membrane traces and c~0:1 otherwise, in both low and high states. Right: Spike train cross-correlation functions for the firing of the two neurons when receiving correlated input, showing state dependent shaping. (B) Relationship betweenspike count correlation rT for windows of length T and input correlation c, showing linearity for small c and a dependence on T . (C) Outputcorrelation as a function of window size in the high and low states. Asterisks mark the values of T that correspond to the plots in Figure 3B. (D) Ratioof correlations as a function of window size in the high and low states, showing favoring of short timescale synchrony in the high state. For

comparison, the lack of correlation shaping for a purely linear neural transfer is indicated. (E) RMS coherence (jCC12(f )j=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCC11(f )CC22(f )

q) between spike

trains showing a decrease in low-frequency coherence and increase in high frequency coherence in the high state. The theoretical results (solid lines)

shown in in panels (B) through (E) were derived from a linear response calculation valid in the small c limit (see Methods). Bars denote standard errorin (B) through (D). In (B), standard error is smaller than the width of the dots.doi:10.1371/journal.pcbi.1002305.g003

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 6 December 2011 | Volume 7 | Issue 12 | e1002305

grals can be transformed to the frequency domain, using the

Wiener-Khinchin theorem [44] to relate correlation functions

Cij(t) to their spectral analogues CCij(f ), yielding

rT~

Ð?{? CCij(f )kT (f )dfÐ?{? CCii(f )kT (f )df :

ð7Þ

Here kT (f ):1

p2Tf 2sin2 2pfT

2

� �is the Fourier transform of the

triangular weighting term in Eq. (6). Our strategy was to relate the

cross spectrum between the spike trains, CCij(f ), to single neuron

integration properties.

Single neuron input-output transfer is typically expressed

through its spectral transfer function AA(f ). The transfer function

measures the ratio of the amplitudes of a neuron’s firing rate

response and a small amplitude sinusoidal signal of frequency f(Figure 4A). For very slow inputs, the transfer function jAA(0)jequals the firing rate gain, since this measures the sensitivity of

firing responses to static (f&0) inputs. For f w0, jAA(f )j is the

susceptibility for a neuron’s trial averaged response to be locked to

a time varying signal. The transfer function jAA(f )j is experimen-

tally measurable [60], and is related to the more commonly

reported spike triggered average [61]. In general, for neurons in

the fluctuation-driven regime, jAA(f )j is a decaying function of f(Figure 4B).

If each neuron receives a small shared signal Q(t), then we can

write the expectation of the Fourier transform of the spike train

from neuron i as:

Syyi(f )T&AAQ(f )QQ(f ), ð8Þ

where the brackets denote an average over repeated frozen

presentations of the shared signal Q(t) with different realizations of

the independent noise driving the neurons [62]. Here, AAQ(f ) is the

linear response of the system to the perturbation Q(t). Finally,

averaging the quantity Syy�1(f )TSyy2(f )T over different realizations

of the process Q(t) yields the cross-spectrum between neurons 1

and [17,62,63,64]:

CC12(f )~SSyy�1(f )TSyy2(f )TTQ&jAAQ(f )j2SQQ�QQT: ð9Þ

For the case of white noise input, we have that SQQ�QQT~cs2. With

Eqs. (7) and (9) we calculated the spike count correlation

coefficient between the two neurons receiving shared white noise

input as

rT&ST c~s2Ð?{? jAAj

2(f )kT (f )dfÐ?

{? CCii(f )kT (f )df

!c: ð10Þ

Our theory then relates single neuron transfer jAA(f )j and power

spectrum CCii(f ) to the joint pairwise response rT .

The theoretical predictions given in Eq. (10) gave a very good

quantitative match to simulations of the leaky integrate-and-fire

neuron pair (Figures 3B–E, compare solid curves to points),

capturing the correlation shaping between the two states. Eq. (10)

has been previously derived [17,18], however, the model neurons

considered in those studies were current driven model neurons.

We considered conductance driven model neurons, meaning that

the calculation of jAA(f )j and CCii(f ) must account for the linked

shifts of the membrane time constant and membrane potential

fluctuations from the low to the high state (Figure 2B). For our

conductance based integrate-and-fire model neurons, the quanti-

ties AA(f ) and CCii(f ) were calculated by numerically integrating the

Fokker-Planck equation associated with the stochastic differential

equation expressed in Eq. (1) (see [42] and Methods). The

distinction between current and conductance based neural

integration will be shown to be critical for correlation shaping.

Before correlation shaping is related to the shifts in AA( f )between the low and high states, we first discuss the dependence of

susceptibility ST on the window size T (Figure 3B). This

dependence enters equation (Eq. (10)) through the weighting term

kT (f ), which determines the contribution of AA(f ) across frequency

to ST . For long timescales (large T ), kT (f ) is low-pass, so that only

the neurons’ response to low frequencies contributes to correlation

susceptibility. In contrast, for short timescales (small T ), kT (f )weighs the transfer function approximately equally across all

frequencies. Hence, the neurons’ high frequency response

determines precise spike synchrony. Indeed, for T?? we have

that kT (f )?d(f ), while T?0 limits kT (f ) to a constant function

on ({?,?). Therefore, for large T , only the zero-frequency

components of AA(f ) contribute to the integral, while for small T ,

all frequencies contribute.

A mechanistic understanding of correlation shaping (Figure 3D)

requires knowledge of how the rate of balanced synaptic activity

affects the transfer function. As discussed previously, the increase

in synaptic input from the low to the high state decreased the

effective membrane time constant of the neuron t while it

increased the input variability s (Figure 2B). The decrease in tcorresponded to a decrease in the timescale over which a neuron

integrates inputs and hence an attenuation of the neuron’s transfer

function. For low frequency inputs, this reduction was precisely the

firing rate gain control known to occur with increased synaptic

input (Figure 2C). Increased variability and shunting due to

heightened conductance reduced the neuron’s ability to respond to

slow depolarizing inputs. However, the reduction in the transfer

function from the low to high state was not uniform across all

frequencies (Figure 4C, Left). This was because the smaller value

of t in the high state enhanced the tracking of fast inputs,

mitigating the attenuation of the transfer function for high

frequencies. The combination of the non-uniform attenuation of

the transfer function and increase in s from the low to high state

determined the shaping of the correlation susceptibility ST (see Eq.

(10)).

To illustrate the shift in single neuron response between the low

and high states, we considered the quantity sjAA(f )j, the strength of

the input fluctuations multiplied by the input transfer function.

The ratio jsAAjhigh=jsAAjlow was an increasing function of frequency

(Figure 4C, Center), indicating that high frequency transfer is

favored in the high state. In general, a favoring of high frequencies

corresponds to a favoring of synchrony, measured over only small

T (since kT (f ) is nearly flat across f for small T ). Thus, the high

state is expected to favor small T correlation transfer compared to

the low state (Figure 4C, Right). In contrast, for large T which

corresponds to low frequencies, correlation transfer was disfavored

in the high state (since kT (f ) only weights low f for large T ). This

ratio allowed us to intuitively link correlation shaping over

different timescales to the shaping of the transfer function over

different frequencies.

We argue above that a change in the effective membrane time

constant is central to the correlation shaping we discuss. To

demonstrate this fact, we computed the transfer function and

correlations for a current-based model in which t remained

unchanged in the low and high state, although s increased by the

same amount. If firing rates were again fixed at 15 Hz, the transfer

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 7 December 2011 | Volume 7 | Issue 12 | e1002305

function was again reduced in the high state, but the ratio

jsAAjhigh=jsAAjlow remained close to unity (Figure 4D, Left and

Center). As a result, no substantial correlation shaping was

observed (Figure 4D, Right). The above comparison shows that

this shaping requires the modulation of cellular properties that is

allowed by a conductance-based model.

Finally, we note that, although our analysis has focused on the

numerator of Eq. (7), the denominator also affects the correlation

for large time windows (Figure S4). For these values of T , the

denominator was increased in the high state, reflecting the higher

variability of firing due to stronger input fluctuations. This further

attenuated the value of rT for large T in the high state.

Correlation Shaping with Different Output Firing RatesTo avoid changes in correlation owing to firing rate [17], we

chose the balance between excitation and inhibition in previous

sections so that firing rate was fixed across both the low and high

states (Figure 2C). However, it is unlikely that firing rates will

remain fixed as a network shifts from a low conductance to a high

conductance state. Thus, it is important to understand how

correlation shaping via balanced excitatory and inhibitory inputs

interacts with the correlation changes expected due to firing rate

changes. In this section we show how the modulations of

correlation due to balanced excitatory and inhibitory inputs and

those due to firing rate changes from imbalanced inputs are

distinct.

The firing rates of our output neurons were determined by the

input rate of both the excitatory (Re) and inhibitory (Ri) inputs. In

fact, for any desired output rate, there was a curve in (Ri,Re) space

that achieved that rate (Figure 5A). For moderate input rates, a

balanced shift in input (approximately linear in Ri and Re)

preserved output firing rate. A change in output firing rate

(switching from one curve to another in Figure 5A), can occur

from a shift in Re, a shift in Ri, or some combination of the two.

When we fixed Ri to its value in the low state and increase Re so

that the output rate increased, rT increased over all timescales T(Figure 5B, top), as expected [17]. A similar effect occured if we

repeat this in the high state (Figure 5B, bottom). Thus, the

modulation of rT by a rate change due to an imbalanced shift of

Re simply scales rT for all T (collapsed blue and orange curves in

Figure 5C). Nevertheless, after correcting for the rate scaling of rT ,

the shaping of correlation between the low and high states

Figure 4. Relating correlation shaping to single neuron transfer (A) Illustration of neuronal transfer function. A perturbing input ofamplitude I and frequency f causes a modulation of the spike response of a fluctuation driven neuron. Averaging across stimulus presentation trialsgives the average output firing rate r(t) with amplitude O. The output-input ratio defines the neural transfer AA(f )~O=I . (B) Example jAA( f )j for afluctuation driven neuron (black curve). The weighting function kT (f ) for T~3 ms and 50 ms (grey curves). (C) Left: transfer function jAA( f )j for neuronsin the low and high background states. Center: Ratio of transfer functions in the two states (normalized by change in input strength shigh=slow). Right:Ratio of correlations as in Figure 3D. (D) Same as (C), but for a current-based model in which teff does not change in the high state. Note correlationshaping in (C), Right but not (D), Right.doi:10.1371/journal.pcbi.1002305.g004

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 8 December 2011 | Volume 7 | Issue 12 | e1002305

remained clear (Figure 5C), demonstrating that correlation

shaping due to a change from low to high states is distinct from

correlation shifts due to arbitrary output firing rate changes.

To illustrate this, we considered a shift from 8 Hz in the low

state to 35 Hz in the high state. In the shift from the low to high

state, the effective membrane timescale teff shifted from 10.8 to

2.9 ms and the amplitude of the input fluctuations s from 0.16 to

0.37 nA. These shifts changed sjAA(f )j significantly (as discussed in

the previous section), and changed the timescales over which the

neuron pair was correlated. This was contrasted by a shift from

8 Hz to 35 Hz in the low state: a change in firing rate without a

change between low and high states. Here, teff shifted from 10.8

to 10.2 ms and the input fluctuations s from .16 to .18 nA, having

little influence on sjAA(f )j other than a uniform scaling due to the

output rate change. In total, by changing both Re and Ri, it was

possible to not only change the output firing rate so as to amplify

or attenuate rT , but also to shape the timescales over which a

neuron pair was correlated.

Experimental Verification with Dynamic ClampRecordings

Our two-neuron framework for studying correlation transfer

(Figure 1A) permited an experimental verification of correlation

shaping with balanced, fluctuating conductance inputs. We

performed in vitro patch clamp recordings from cortical pyramidal

neurons receiving simulated excitatory and inhibitory inputs.

Unlike past experimental studies of correlation transfer [16,17],

our model involved conductance-based, rather than current-based

synapses. Therefore, we simulated synaptic input using dynamic

clamp [65] (see Methods), which affected the membrane

integration timescale as well as membrane potential variability.

We chose maximal excitatory and inhibitory conductances of 1 nS

and synaptic timescales of 6 and 8 ms, respectively, producing a

synaptic input that was more biophysically realistic than the

diffusion process used in previous sections (Figure 6A). The shift

from low to high state caused a near two-fold reduction in firing

rate gain (Figure 6B), in qualitative agreement with our model

simulations (Figure 2C) and past dynamic clamp studies [29].

Further, as was done in the model, we set the synaptic balance in

the low and high states to produce approximately the same firing

rate (5:5+:9 Hz in the low state and 6:0+1:5 Hz in the high

state).

The correlated input for a given neuron pair was a mixture of

shared and independent excitatory and inhibitory inputs, mim-

icking the input provided to the model (Figure 3A). The partial

overlap in the synaptic input produced correlated membrane

potential and spike dynamics for every neuron pair in both the low

and high states. Our recorded spike trains showed a dependence of

spike count correlation on T that was qualitatively similar to that

of the model, apparent in the ratio of rT in the high and low states

(Figure 6C). The ratio was a decreasing function of T , indicating a

bias toward synchrony in the high state compared to the low state.

This shape was consistent with our model results (Figure 3D),

although the ratio did not fall substantially below unity in the limit

of large T . This suggested that the decrease in gain jAA(0)j and the

increase in variability s from the low to high state were of similar

magnitudes, since in the limit of large T correlation susceptibility is

proportional to s2jAA(0)j2 (Methods). A conductance-based

simulation using the same synaptic parameters used for dynamic

clamp stimulation produced results in agreement with the

experiment (Figure S5). The favoring of synchrony (T = 2 ms)

over long timescale correlation (T = 200 ms) in the high state was

statistically significant in a pairwise analysis across the dataset

(Figure 6C, inset; Pv3|10{5, paired t-test). The experiments

demonstrated that an increase in the rate of balanced conductance

input shapes pairwise correlation so as to favor synchronization

over long timescale correlation, thereby verifying the main

theoretical predictions of our study.

Our theoretical treatment has ignored the timescale of synaptic

input, and has associated all filtering to the membrane and spike

properties of the model (Figure 4). Correlation transfer with

realistic synaptic timescales did quantitatively differ from the case

with instantaneous synaptic input (Figure S5B). Nevertheless, our

theoretical work captured the main effects of correlation shaping

when synaptic timescales were realistic (Figures 6 and S5). This is

because only the effective membrane time constant was sensitive to

a shift in input firing rate, which our theory accounts for, while

synaptic filtering did not change between low and high states. We

Figure 5. Comparing correlation shaping due to balanced excitatory and inhibitory inputs and correlation shifts due to shift inoutput rate. (A) Output firing rate as a function of Ri and Re (firing rates above 100 Hz not shown). The curves in the space that yield output firingrates of 8,15, and 35 Hz are labeled. The lines lie in a small region of the full space, corresponding to the region where excitation and inhibition arebalanced. (B) Top: Spike count correlation as a function of T for the three output rates, where rate changes are due to a change Re , with Ri fixed atthe low state value. Bottom: same as top, except that Ri is fixed at the high state value. (C) The curves in B for the low and high states scaled to matchthe center curve (15 Hz) at T~100 ms.doi:10.1371/journal.pcbi.1002305.g005

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 9 December 2011 | Volume 7 | Issue 12 | e1002305

remark that, for synapses with very long timescales, correlation

shaping should only be present for large T , since correlations at

small T will be negligible.

Consequences of Correlation Shaping for SignalPropagation

The spike train correlations between neuron pairs substantially

influence the propagation of neural activity in feedforward

architectures [13]. For example, while our study has so far

focused on the transfer of correlation for neuron pairs receiving

common input, the firing rate of a single downstream neuron also

depends on the correlation between neurons in its pre-synaptic

pool [12]. If the integration timescale of the downstream target is

small, only precise spike synchrony will effectively drive the

neuron. In contrast, neurons that slowly integrate inputs will be

sensitive to long timescale correlations. In our study, we

demonstrated that an increase in the rate of synaptic input

increases spike count correlation at small T while simultaneously

decreasing the correlation at large T (Figure 3D). We therefore

expected that this correlation shaping would influence the extent

to which activity can be propagated to a downstream layer.

Further, that the magnitude of this effect would depend on the

integration timescales of the downstream targets.

As an illustration of this effect in a simplified system, we studied

the firing rate of a downstream neuron receiving input from an

upstream population of correlated neurons (Figure 7A; Methods).

The level of synaptic drive from layer 1 shaped the correlation of

pairs of layer 2 neurons (Figure 7A, insets). The network was

constructed so that the activity of any given pair of neurons in

Layer 2 was equivalent to that of the neuron pairs studied in

previous sections. As the correlation of layer 2 spike outputs was

shaped, so too was the magnitude and timescale of the synaptic

drive to the downstream target neuron (Figure 7B). For

comparison, we show that downstream target’s synaptic input

when the layer 2 neurons were uncorrelated (Figure 7B, bottom),

showing significantly reduced variability [12]. In the uncorrelated

case, the firing rate of the downstream target was much less than

1 Hz, indicating that correlated input was necessary for its

recruitment.

We study how correlation shaping of the layer 2 projections

affected the recruitment of the downstream target neuron. In

particular, we focused on how the changing timescale of

correlation recruited downstream targets differentially, depending

on their own integration properties. We varied the rate of

balanced synaptic input from layer 1 to layer 2 in a smooth

manner (following the Re and Ri path for 15 Hz output in

Figure 5A), gradually shaping the correlation function between

any given layer 2 neuron pair. The shaping included the low and

high states described earlier as near endpoints on a continuum

(Figure 7C). When the downstream target had a smaller time

constant (3 ms), its firing rate was increased when the pre-synaptic

population was in the high state (Figure 7C, dashed line). This

Figure 6. Correlation shaping in cortical dynamic clamp experiments. (A) Left: Recorded average EPSPs and IPSPs from resting neuronsshowing membrane voltage (Vm) deflections, with corresponding conductances ge,i . Right: Voltage traces from example recorded neuron pairs inhigh and low states. The degree of synaptic overlap was 0.5 for both high and low states. The inter-spike interval coefficient of variation increasedfrom 0.40 in the low state to 0.48 in the high state. (B) Firing rate versus mean input current curves for neurons in low and high states showingreduction in gain in the high background state. (C) The ratio of rT in the high to low state as a function of window size T (compare to Figure 3D).Curves are population average results (n = 28) with the shaded region denoting the standard error. Inset: Correlation ratio shown at T = 2 ms and200 ms for each recorded pair.doi:10.1371/journal.pcbi.1002305.g006

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 10 December 2011 | Volume 7 | Issue 12 | e1002305

contrasted with the decreased firing rate in the high state when the

downstream target had a longer time constant (20 ms) (Figure 7C,

solid line). This differential effect was due to matching between the

correlation timescale of layer 2 and the integration timescale of the

downstream target. In the high state, synchrony drove the neuron

with the short integration timescale, while, in the low state, long

timescale correlations drove the slower neuron. Note that the

firing rate of layer 2 neurons was unchanged in all cases studied.

This simple example demonstrates that the structure of correla-

tions between pre-synaptic neuron pairs can differentially drive

downstream targets depending on their integration properties.

Discussion

We have demonstrated that the rate of balanced synaptic input

changes the correlation timescale of spike trains of a pair of neurons

receiving partially correlated input. High rate synaptic input promoted

precise spike time synchrony, while low rate synaptic input enhanced

long timescale correlation. This correlation shaping was independent

of changes in input correlation or the output firing rate of the neuron

pair. Rather, it required a thresholding nonlinearity between input

and spike train response as well as a state-dependent integration

timescale. Both of these are properties of many neurons in the central

nervous system, and hence we expect that similar correlation shaping

may occur in a variety of brain regions.

Correlation Shaping Compared to Other Forms ofCorrelation Modulation

Correlated neural activity continues to receive increasing

attention [1], prompting investigations of the mechanisms that

determine the transfer of correlation. Correlations are typically

measured only at one timescale, but as we have shown, the

magnitude of correlation depends on the timescale being

considered, as does the likely significance of this correlation for

activation of downstream neurons. Past studies have highlighted

the dependence of spike train correlations on the magnitude of

input correlation [15,16], the form of spike excitability [19,66], or

the firing rate of the neuron pair [17,18]. However, how the

timescale of correlations are modulated through plausible

mechanisms had not been addressed. Changes in membrane

conductance have been widely studied and strongly influence the

dynamics of single neuron activity [22]. In our study, we found

that timescale-specific changes in neural correlations are a

necessary consequence of conductance based modulation schemes.

Previous work that has examined how correlated activity is

transferred has used linear response methods to examine the

response of neurons to current fluctuations, thereby leaving

membrane integration invariant [17,18]. As a result, cellular

properties such as timescale were not modulated (see Figure 4D).

We showed that when synaptic conductance is considered, it is

possible to shape both the magnitude and timescale of output spike

Figure 7. Effects of correlation shaping on the propagation of neural activity. (A) Schematic of layered network. Layer 1 neurons aremodeled as Poisson processes and are either in the low (top) or high (bottom) state. Each layer 2 neuron receives a combination of private input anda globally common input from layer 1. The common input correlates each pair of layer 2 neurons, while the state of layer 1 shapes the correlations(cross correlation function insets). The layer 2 neurons have marginal and pairwise statistics identical to the neurons in Figures 2 and 3. The spikeoutputs of the layer 2 neurons converge onto a downstream neuron with integration timescale tdownstream. (B) Example realization of the summedsynaptic activity that drives the downstream target neuron in the low (top), high (middle), and, for comparison, when the layer 2 neurons areuncorrelated (bottom). (C) Effect of the state change on the downstream neuron’s firing rate. The horizontal axis Re shows the level of excitatorysynaptic activity that the neurons in the second layer received from the first layer. Ri is adjusted in a balanced fashion so that the layer 2 neurons fireat 15 Hz (see Figure 5A). The downstream target neuron has either tdownstream~3 ms or tdownstream~20 ms. The neuron with the fast time constantwas driven more strongly in the high state. However, the neuron with the slow time constant showed a decreased firing rate in the high state.doi:10.1371/journal.pcbi.1002305.g007

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 11 December 2011 | Volume 7 | Issue 12 | e1002305

train correlations. This is a novel result that is nevertheless

consistent with, and complementary to, the observation that firing

rate also modulates correlations (see Figure 5).

Noise Correlation Shaping in Neural CircuitsThe widespread use of multi-unit recording techniques to study

population activity has produced an increasingly clear picture of

how neuronal spike trains are correlated in a variety of neural

states. Recently, there has been particular interest in noise

correlations, which are specific to within trial comparisons and

cannot be directly attributed to a common signal [10].

Several groups have reported noise correlation measurements,

ranging from small positive values [58,2,3,6,7,8] to values that are,

on average, zero, with positive and negative values equally

represented [4,67,48]. Furthermore, in cases where significant

noise correlation is measured, it can be modulated on distinct

timescales. In the visual system, for example, noise correlation

measured on timescales less than 100 ms is largest for cells with

similar preferred stimulus orientations being driven at that

orientation, observed in both spike responses [2] and synaptic

input [37]. Further, while increasing stimulus contrast enhances

short timescale correlation, it reduces long timescale (w100 ms)

correlation [2]. In primate area V4, stimulus attention reduces

noise correlation when measured on timescales that are larger than

100 ms, yet has little influence on short timescale correlation [7,8].

In contrast, other groups have shown that stimulus attention

enhances spike synchrony measured at the gamma frequency

timescale (20–40 ms) [68]. In the electrosensory system, long

timescale noise correlation is reduced by recruitment of a non-

classical receptive field, while synchrony is increased under the

same conditions [3]. Thus, spike train noise correlations provide

an excellent framework to study how the magnitude and timescale

of correlations are shaped by neural state changes.

While a shaping of output correlation observed in these systems

may be inherited from a state-dependence of input correlation

(Figure 1B), single neuron response properties are often also

modulated by network state. This suggests that a shift in

correlation susceptibility may underlie a shift in pairwise

correlation (as in Figure 1C). Indeed, firing rate gain is modulated

by attention [69], stimulus contrast [21], and the recruitment of a

non-classical receptive field [70]. In many cases, intracellular

recordings have established that gain control is mediated by an

increase in the rate of excitatory and inhibitory synaptic inputs

[21,31], in a fashion similar to the case presented in our study.

Dual intracellular experiments that measure both input and output

correlation across distinct neural states [45,46,37] are required to

parcel the contribution of correlation inheritance and correlation

transfer to the full shift in noise correlations.

Connecting Single Neuron and Network ModulationsA central result of our paper is that changes in synaptic input

rate shape the correlation between the output spike trains from a

pair of neurons. This is a consequence of how synaptic input

modulates the timescale of membrane integration and response

sensitivity of the two neurons. Our theoretical analysis formalizes

this concept by explicitly relating the spike train correlation

coefficient to the single neuron transfer function. Though we

focused on modulation by balanced synaptic inputs, the relation-

ship between transfer function and correlation is general, requiring

only that the input correlation be sufficiently small. Thus, we

predict that any synaptic or cellular mechanism that modulates

single neuron transfer will necessarily affect spike train correla-

tions.

Modulation of single neuron transfer with the level of synaptic

input rate is well studied [27,28,29,30,31,32]. However, how other

cellular processes affect neuronal transfer is equally well studied.

For example, increases in the spike after-hyperpolarization [50] or

decreases in the spike after-depolarization [51] reduce the gain of

the firing rate response to static driving inputs. Sustained firing

often recruits slowly activating adaptation currents that also reduce

gain [52,53]. We predict that these modulations will reduce long

timescale spike rate correlations. In contrast, the presence of low

threshold potassium currents in the auditory brainstem [71]

promotes high frequency single neuron transfer and thus may also

promote pairwise synchronization. In total, our result gives a

general theory that links the modulation of single neuron and

network responses, thereby expanding the applicability of studies

of single neuron modulation.

Selective Propagation of Neural ActivityHow the brain selectively propagates signals is a basic question

in systems neuroscience. One control mechanism is through an

‘unbalancing’ of feedforward excitation to inhibition, with

disinhibited populations propagating activity and excessive

inhibition silencing propagation [72]. Modulation of correlation

is an alternative mechanism to control signal propagation. The

correlation between spike trains from neurons in a population

enhances the ability of that population’s activity to drive

downstream targets [12,13]. We have shown that modulating

the timescale of correlation in the upstream population to match

the integration timescale of the downstream population improves

signal propagation (Figure 7). Matching the integration dynamics

of distinct neuronal populations to one another is a common

theme in the binding of distributed activity [73]. In previous

studies, the phase relationship between distinct neuronal popula-

tions both oscillating at some frequency gated the interaction

between distinct brain regions. Our study did not assume rhythmic

population dynamics, but rather only matched integration

timescales.

The nonlinearity of spike generation allows for the transfer of

shared input to multiple neurons to be controlled in complex ways.

We have shown that well-studied mechanisms of single neuron

response modulation, such as firing rate gain control, have direct

relations to changes in correlation for neuron pairs. Thus, state

dependent shifts in single neuron transfer also influence how

populations of neurons coordinate their activity. Our results are a

step in understanding how the collective behavior of neuronal

networks can be controlled in different brain states.

Supporting Information

Figure S1 Diffusion limit shows qualitative effects of correlation

shaping. (A) Top: Correlation in the low and high states for a

conductance-based model with alpha-function synapses. The

excitatory time constant was 2.5 ms and the inhibitory time

constant 5 ms. The amplitude of the alpha function was taken so

that it matched with the delta-function synapses described in the

main text. Other parameters were as in the main text. Bottom:

Ratio of correlations between the high and low states. (B) Same as

(A), but after taking the diffusion approximation (see Eq. 1 in the

main text). (C) Same as (B), but after taking s(V )~s(Eeff ). The

ratio rThigh=rT

low exhibits similar correlation shaping in all cases.

(PDF)

Figure S2 Comparison between simulation and experimental

results. (A) Top: Correlation in the low and high states calculated

from dynamic clamp experiments. Bottom: Ratio of correlations

between the high and low states. (B) Similar to (A), showing results

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 12 December 2011 | Volume 7 | Issue 12 | e1002305

from a conductance-based model with alpha-function synapses.

The excitatory time constant was 6 ms and the inhibitory time

constant 8 ms. The firing rate was 5 Hz to match experiments.

Other parameters were as in the main text.

(PDF)

Figure S3 Results hold for large c. Top: Correlation in low and

high states for c~0:5, parameters otherwise identical to Figure 3 in

the main text. Bottom: Ratio of correlations in the low and high states.

(PDF)

Figure S4 Change in power spectrum of the spike train y(t) from

low to high states. In both cases, the high-frequency limit of the power

spectrum is equal to the firing rate of the neuron. For low frequencies,

however, the power was increased in the high state, reflecting the

increased variability of firing in the high state (note that as

frequency?0, the power spectrum is equal to the firing rate

multiplied by the square of the inter-spike interval CV). To determine

the denominator of Eq. (7), we integrate the power spectrum by kT (f )to obtain Var(nT ). When T is small, Var(nT ) is identical in the two

states, because the high frequency limits of the power spectrum are

equal. When T is large, Var(nT ) is increased in the high state,

because the low frequency limit of the power spectrum is enhanced.

(PDF)

Figure S5 Correlation shaping occurs for different synaptic

strengths. (A) Theoretically calculated correlation curve for

ae~:02, ai~:04, Re~1 kHz in the low state and 4.08 kHz in

the high state. The time constant decreased from 7.5 ms in the low

state to 1.9 ms in the high state. Firing rates were 15 Hz in both

states. (B) Theoretically calculated correlation curve for ae~:005,

ai~:01, Re~2 kHz in the low state and 8 kHz in the high state.

The time constant decreased from 15.2 ms in the low state to

4.8 ms in the high state. Firing rates were 15 Hz in both states.

(PDF)

Acknowledgments

We thank Thanos Tzounopoulos for comments on the manuscript,

Magnus Richardson for discussions concerning numerical techniques, and

Rick Gerkin for developing the dynamic clamp software.

Author Contributions

Conceived and designed the experiments: ALK AMO NNU BD.

Performed the experiments: ALK AMO. Analyzed the data: ALK AMO

BD. Wrote the paper: ALK BD.

References

1. Cohen M, Kohn A (2011) Measuring and interpreting neuronal correlations.Nat Neurosci 14: 811–819.

2. Kohn A, Smith MA (2005) Stimulus dependence of neuronal correlation inprimary visual cortex of the macaque. J Neurosci 25: 3661–3673.

3. Chacron MJ, Bastian J (2008) Population coding by electrosensory neurons.

J Neurophysiol 99: 1825–1835.4. Greenberg DS, Houweling AR, Kerr JND (2008) Population imaging of ongoing

neuronal activity in the visual cortex of awake rats. Nat Neurosci 11: 749–751.

5. Kohn A, Zandvakili A, Smith MA (2009) Correlations and brain states: fromelectrophysiology to functional imaging. Curr Opin Neurobiol 19: 434–438.

6. Gutnisky DA, Dragoi V (2008) Adaptive coding of visual information in neuralpopulations. Nature 452: 220–224.

7. Cohen MR, Maunsell JHR (2009) Attention improves performance primarily by

reducing interneuronal correlations. Nat Neurosci 12: 1594–1600.8. Mitchell JF, Sundberg KA, Reynolds JH (2009) Spatial attention decorrelates

intrinsic activity uctuations in macaque area V4. Neuron 63: 879–888.

9. Vaadia E, Haalman I, Abeles M, Bergman H, Prut Y, et al. (1995) Dynamics ofneuronal interactions in monkey cortex in relation to behavioural events. Nature

373: 515–518.10. Averbeck BB, Latham PE, Pouget A (2006) Neural correlations, population

coding and computation. Nat Rev Neurosci 7: 358–366.

11. Harris KD (2005) Neural signatures of cell assembly organization. Nat RevNeurosci 6: 399–407.

12. Salinas E, Sejnowski TJ (2001) Correlated neuronal activity and the ow of neural

information. Nat Rev Neurosci 2: 539–550.13. Kumar A, Rotter S, Aertsen A (2010) Spiking activity propagation in neuronal

networks: reconciling different perspectives on neural coding. Nat Rev Neurosci11: 615–627.

14. Ginzburg I, Sompolinsky H (1994) Theory of correlations in stochastic neural

networks. Phys Rev E 50: 3171.15. Moreno-Bote R, Parga N (2006) Auto-and crosscorrelograms for the spike

response of leaky integrate-and-fire neurons with slow synapses. Phys Rev Lett96: 28101.

16. Galan RF, Fourcaud-Trocme N, Ermentrout GB, Urban NN (2006)

Correlation-Induced synchronization of oscillations in olfactory bulb neurons.J Neurosci 26: 3646–3655.

17. de la Rocha J, Doiron B, Shea-Brown E, Josic K, Reyes A (2007) Correlation

between neural spike trains increases with firing rate. Nature 448: 802–806.18. Shea-Brown E, Josic K, de la Rocha J, Doiron B (2008) Correlation and

synchrony transfer in Integrate-and-Fire neurons: Basic properties andconsequences for coding. Phys Rev Lett 100: 108102.

19. Marella S, Ermentrout GB (2008) Class-II neurons display a higher degree of

stochastic synchronization than class-I neurons. Phys Rev E 77: 041918.20. Ostojic S, Brunel N, Hakim V (2009) How connectivity, background activity,

and synaptic properties shape the Cross-Correlation between spike trains.J Neurosci 29: 10234–10253.

21. Anderson JS, Lampl I, Gillespie DC, Ferster D (2000) The contribution of noise

to contrast invariance of orientation tuning in cat visual cortex. Science 290:1968–1972.

22. Destexhe A, Rudolph M, Pare D (2003) The high-conductance state of

neocortical neurons in vivo. Nat Rev Neurosci 4: 739–751.

23. Shadlen MN, Newsome WT (1994) Noise, neural codes and corticalorganization. Curr Op Neurobiol 4: 569–579.

24. van Vreeswijk C, Sompolinsky H (1996) Chaos in neuronal networks with

balanced excitatory and inhibitory activity. Science 274: 1724–1726.

25. Heiss JE, Katz Y, Ganmor E, Lampl I (2008) Shift in the balance betweenexcitation and inhibition during sensory adaptation of s1 neurons. J Neurosci 28:

13320–13330.

26. Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrentbalanced cortical activity. Nature 423: 288–293.

27. Ho N, Destexhe A (2000) Synaptic background activity enhances the

responsiveness of neocortical pyramidal neurons. J Neurophysiol 84: 1488–1496.

28. Doiron B, Longtin A, Berman N, Maler L (2001) Subtractive and divisiveinhibition: Effect of Voltage-Dependent inhibitory conductances and noise.

Neural Comput 13: 227–248.

29. Chance FS, Abbott L, Reyes AD (2002) Gain modulation from backgroundsynaptic input. Neuron 35: 773–782.

30. Mitchell SJ, Silver R (2003) Shunting inhibition modulates neuronal gain during

synaptic excitation. Neuron 38: 433–445.

31. Cardin JA, Palmer LA, Contreras D (2008) Cellular mechanisms underlyingstimulusdependent gain modulation in primary visual cortex neurons in vivo.

Neuron 59: 150–160.

32. Ly C, Doiron B (2009) Divisive gain modulation with dynamic stimuli in

integrate-and-fire neurons. PLoS Comput Biol 5: e1000365.

33. Salinas E, Sejnowski TJ (2000) Impact of correlated synaptic input on output

firing rate and variability in simple neuronal models. J Neurosci 20: 6193–6209.

34. Moreno R, de La Rocha J, Renart A, Parga N (2002) Response of spikingneurons to correlated inputs. Phys Rev Lett 89: 288101.

35. Rudolph M, Destexhe A (2001) Correlation detection and resonance in neural

systems with distributed noise sources. Phys Rev Lett 86: 3662–3665.

36. Cafaro J, Rieke F (2010) Noise correlations improve response fidelity andstimulus encoding. Nature 468: 964–967.

37. Yu J, Ferster D (2010) Membrane Potential Synchrony in Primary Visual Cortex

during Sensory Stimulation. Neuron 68: 1187–1201.

38. Giridhar S, Doiron B, Urban N (2011) Timescale-dependent shaping ofcorrelation by olfactory bulb lateral inhibition. Proc Natl Acad Sci USA 108:

5843.

39. Koch C (1999) Biophysics of computation: information processing in single

neurons, volume 428. New York: Oxford University Press.

40. Richardson M (2004) Effects of synaptic conductance on the voltage distribution

and firing rate of spiking neurons. Phys Rev E 69: 051918.

41. Richardson M, GerstnerW (2005) Synaptic shot noise and conductance

uctuations affect the membrane voltage with equal significance. Neural Comput17: 923–947.

42. Richardson M (2008) Spike-train spectra and network response functions for

non-linear integrate-and-fire neurons. Biol Cybern 99: 381–392.

43. Risken H (1996) The Fokker-Planck Equation: Methods of Solutions and

Applications. Springer.

44. Cox D, Isham V (1980) Point Processes. London: Chapman and Hall.

45. Poulet JFA, Petersen CCH (2008) Internal brain state regulates membranepotential synchrony in barrel cortex of behaving mice. Nature 454: 881–885.

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 13 December 2011 | Volume 7 | Issue 12 | e1002305

46. Okun M, Lampl I (2008) Instantaneous correlation of excitation and inhibition

during ongoing and sensory-evoked activities. Nat Neurosci 11: 535–537.

47. Trong PK, Rieke F (2008) Origin of correlated activity between parasol retinal

ganglion cells. Nat Neurosci 11: 1343–1351.

48. Renart A, de la Rocha J, Bartho P, Hollender L, Parga N, et al. (2010) The

asynchronous state in cortical circuits. Science 327: 587–590.

49. Moreno-Bote R, Parga N (2006) Auto- and crosscorrelograms for the spike

response of leaky integrate-and-fire neurons with slow synapses. Phys Rev Lett

96: 028101.

50. Higgs MH, Slee SJ, Spain WJ (2006) Diversity of gain modulation by noise in

neocortical neurons: Regulation by the slow afterhyperpolarization conductance.

J Neurosci 26: 8787–8799.

51. Mehaffey WH, Doiron B, Maler L, Turner RW (2005) Deterministic

multiplicative gain control with active dendrites. J Neurosci 25: 9968–9977.

52. Ermentrout B (1998) Linearization of F-I curves by adaptation. Neural Comput

10: 1721–1729.

53. Fernandez FR, White JA (2010) Gain control in CA1 pyramidal cells using

changes in somatic conductance. J Neurosci 30: 230–241.

54. Doiron B, Zhao Y, Tzounopoulos T (2011) Combined ltp and ltd of modulatory

inputs controls neuronal processing of primary sensory inputs. J Neurosci 31:

10579–10591.

55. Shu Y, Hasenstaub A, Badoual M, Bal T, McCormick DA (2003) Barrages of

synaptic activity control the gain and sensitivity of cortical neurons. J Neurosci

23: 10388–10401.

56. Hasenstaub A, Sachdev RNS, McCormick DA (2007) State changes rapidly

modulate cortical neuronal responsiveness. J Neurosci 27: 9607–9622.

57. Burkitt A (2001) Balanced neurons: analysis of leaky integrate-and-re neurons

with reversal potential. Biol Cybern 85: 247–255.

58. Zohary E, Shadlen MN, Newsome WT (1994) Correlated neuronal discharge

rate and its implications for psychophysical performance. Nature 370: 140–143.

59. Kass RE, Ventura V (2006) Spike count correlation increases with length of time

interval in the presence of Trial-to-Trial variation. Neural Comput 18:

2583–2591.

60. Kondgen H, Geisler C, Fusi S, Wang X, Lscher H, et al. (2008) The dynamical

response properties of neocortical neurons to temporally modulated noisy inputsin vitro. Cereb Cortex 18: 2086–2097.

61. Dayan P, Abbott LF (2001) Theoretical Neuroscience: Computational and

Mathematical Modeling of Neural Systems. 1st edition. CambridgeMA: TheMIT Press.

62. Vilela R, Lindner B (2009) A comparative study of different integrate & fireneurons: spontaneous activity, dynamical response, and stimulus-induced

correlation. Phys Rev E 80: 031909.

63. Doiron B, Lindner B, Longtin A, Maler L, Bastian J (2004) Oscillatory activity inelectrosensory neurons increases with the spatial correlation of the stochastic

input stimulus. Phys Rev Lett 93: 048101.64. Lindner B, Doiron B, Longtin A (2005) Theory of oscillatory firing induced by

spatially correlated noise and delayed inhibitory feedback. Phys Rev E 72: 061919.65. Prinz AA, Abbott LF, Marder E (2004) The dynamic clamp comes of age.

Trends Neurosci 27: 218–224.

66. Barreiro A, Shea-Brown E, Thilo E (2010) Time scales of spike-train correlationfor neural oscillators with common drive. Phys Rev E 81: 11916.

67. Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK, et al. (2010)Decorrelated neuronal firing in cortical microcircuits. Science 327: 584–587.

68. Fries P, Reynolds JH, Rorie AE, Desimone R (2001) Modulation of oscillatory

neuronal synchronization by selective visual attention. Science 291: 1560–1563.69. Reynolds JH, Heeger DJ (2009) The normalization model of attention. Neuron

61: 168–185.70. Chacron MJ, Doiron B, Maler L, Longtin A, Bastian J (2003) Non-classical

receptive field mediates switch in a sensory neuron’s frequency tuning. Nature423: 77–81.

71. Slee SJ, Higgs MH, Fairhall AL, Spain WJ (2005) Two-Dimensional time coding

in the auditory brainstem. J Neurosci 25: 9978–9988.72. Vogels TP, Abbott LF (2009) Gating multiple signals through detailed balance of

excitation and inhibition in spiking networks. Nat Neurosci 12: 483–491.73. Womelsdorf T, Schoffelen J, Oostenveld R, Singer W, Desimone R, et al. (2007)

Modulation of neuronal interactions through neuronal synchronization. Science

316: 1609–1612.

Correlation Shaping with Balanced Input

PLoS Computational Biology | www.ploscompbiol.org 14 December 2011 | Volume 7 | Issue 12 | e1002305